Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 492589, 2 pages
doi:10.1155/2012/492589
Editorial
Well-Posed and Ill-Posed Boundary Value
Problems for PDE
Allaberen Ashyralyev,1 Sergey Piskarev,2 Valery Covachev,3
Ravshan Ashurov,4 and Hasan Ali Yurtsever5
1
Department of Mathematics, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey
Science Research Computer Center of Moscow State University, Vorobjevy Gory, Moscow 119899, Russia
3
Department of Mathematics and Statistics, College of Science, Sultan Qaboos University,
123 Muscat, Oman
4
Universiti Putra Malaysia, Institute of Advanced Technology, 43400 Serdang, Selangor, Malaysia
5
American Islamic College, Chicago, IL 60613, USA
2
Correspondence should be addressed to Allaberen Ashyralyev, aashyr@fatih.edu.tr
Received 14 October 2012; Accepted 14 October 2012
Copyright q 2012 Allaberen Ashyralyev et al. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The studies of well-posed and ill-posed local and nonlocal boundary value problems for
partial differential equations are driven not only by a theoretical interest but also by the fact
that several phenomena in engineering, various fields of physics and financial mathematics
can be modeled and investigated in this way.
The present special issue is devoted to the publication of high-quality research
papers in the fields of the construction and investigation of analytic and numerical methods
for solutions of well-posed and ill-posed boundary value problems for partial differential
equations.
The issue covers a wide variety of problems for different classes of ordinary and
partial differential equations, as well as dynamic equations on time scales. The topics
discussed in the contributed papers are traditional for qualitative theory of differential
equations. The issue contains papers on the global well-posedness of the viscous twocomponent Camassa-Holm system, local and global existence of solutions for a generalized
Camassa-Holm equation, global solutions for the Cauchy problem of a Boussinesq-type
equation, exact asymptotic expansion of singular solutions for the 2 1-D Protter problem,
on the regularity for variational evolution integrodifferential in equalities, right-hand side
identification problem arising in biofluid mechanics, regularized solutions of optimal control
problem in a hyperbolic system, generalized localization of Fourier inversion associated with
an elliptic operator for distributions and Kamenev-type oscillation criteria for the secondorder nonlinear dynamic equations with damping on-time scales. Furthermore, classification
2
Abstract and Applied Analysis
of exact solutions for some nonlinear partial differential equations with generalized evolution
is presented.
A number of papers are concerned with well-posedness of partial differential
and difference equations. Interesting stability results are obtained for nonlocal boundary
value problems for hyperbolic Schrödinger equations, difference schemes of ultraparabolic
equations and reverse parabolic problem, approximate solutions of delay parabolic equations
with the Dirichlet condition, difference problem of obtaining the parameter of a parabolic
equation, integral-differential equation of the parabolic type in a Banach space, the first and
second order of accuracy stable implicit different schemes for elliptic-parabolic equations
in Hölder spaces, the second order of accuracy stable difference schemes for the nonlocal
boundary value hyperbolic problem, and the numerical solution of the Bitsadze-Samarskii
nonlocal boundary value problems with the Dirichlet-Neumann condition. Moreover,
applications of generalizations of Wendroff integral inequalities and their discrete analogues
and operator approach to investigate for stability of hyperbolic equations are presented. Two
papers collected in this special issue address construction and investigation of difference
schemes for numerical solutions of stochastic parabolic and hyperbolic equations.
Several authors deal with different aspects of the theory of boundary value problems
for fractional ordinary and partial differential equations. Interesting existence and uniqueness
results are obtained for a class of nonlinear fractional differential equations, system of
nonlinear fractional differential equations with nonlocal and integral boundary conditions,
fractional partial differential equations of higher order. Moreover, applications of operator
approach to investigate for stability of difference schemes for fractional parabolic partial
differential equations with the Dirichlet-Neumann conditions and well-posedness of the
inverse problem for a fractional parabolic equation are presented.
Two papers collected in this special issue address spectrum of differential operators
and its applications for nonlinear Euler buckling problem and asymptotic solutions of
singular perturbed problems with an unstable spectrum of the limiting operator. Moreover,
basis properties of eigenfunctions of second-order differential operators with involution, and
spectral properties of non-self-adjoint perturbations for a spectral problem with involution
are presented. Finally, some applied problems are also considered—an efficient variational
approach for deformable registration of images, and an inverse scattering from a sound-hard
crack via two-step method.
This volume was a collection of 36 accepted manuscripts by 61 authors. The selection
of the papers included in this volume was based on an international peer-review procedure.
The accepted manuscripts examine wide ranging and cutting edge developments in various
areas of well-posed and ill-posed local and nonlocal boundary value problems for partial
differential equations. The papers give a taste of current research. We feel the variety of topics
will be of interest to both graduate students and researchers.
Further, we are very grateful to all authors for sending their new papers for the
publication in the present special issue.
Allaberen Ashyralyev
Sergey Piskarev
Valery Covachev
Ravshan Ashurov
Hasan Ali Yurtsever
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